For sustained nuclear fusion to be possible,
the Lawson criterion must be met,
from which some required properties
of the plasma and the reactor chamber can be deduced.
Suppose that a reactor generates a given power by nuclear fusion,
but that it leaks energy at a rate in an unusable way.
If an auxiliary input power sustains the fusion reaction,
then the following inequality must be satisfied
in order to have harvestable energy:
We can rewrite using the definition
of the energy gain factor ,
which is the ratio of the output and input powers of the fusion reaction:
Returning to the inequality, we can thus rearrange its right-hand side as follows:
We assume that the plasma has equal species densities ,
so its total density .
Then is as follows,
where is the frequency
with which a given ion collides with other ions,
and is the energy released by a single fusion reaction:
Where is the mean product
of the velocity and the collision cross-section .
Furthermore, assuming that both species have the same temperature ,
the total energy density of the plasma is given by:
Where is Boltzmann’s constant.
From this, we can define the confinement time
as the characteristic lifetime of energy in the reactor, before leakage.
Inserting these new expressions for and
into the inequality, we arrive at:
This can be rearranged to the form below,
which is the original Lawson criterion:
However, it turns out that the highest fusion power density
is reached when is at the minimum of .
Therefore, we multiply by to get the Lawson triple product:
For some reason,
it is often assumed that the fusion is infinitely profitable ,
in which case the criterion reduces to:
- M. Salewski, A.H. Nielsen,
Plasma physics: lecture notes,